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**Prompt:** Write an equation for the function graphed below. **Graph Description:** The graph shows a rational function with two vertical asymptotes and no horizontal or slant asymptote visible in the given range. The vertical asymptotes are marked with dashed red lines. 1. **Vertical Asymptotes:** - At \(x = -3\) - At \(x = 3\) 2. **Graph Behavior:** - The function approaches infinity as it nears \(x = -3\) from the left and approaches negative infinity as it approaches from the right. - Similarly, the function approaches negative infinity as it nears \(x = 3\) from the left and approaches infinity as it approaches from the right. - Between the asymptotes \(x = -3\) and \(x = 3\), the function has a distinct behavior, moving from negative to positive. The graph is likely a rational function with factors in the denominator that create the vertical asymptotes at \(x = -3\) and \(x = 3\). **Equation Input Box:** There is a box below the graph for inputting the equation of the function. Based on the graph and the location of the asymptotes, the function may have factors of \((x + 3)\) and \((x - 3)\) in its denominator. Consider starting with an assumption of form for the rational function and adjusting based on any other discernible points or behaviors in the original graph.

**Prompt:** Write an equation for the function graphed below. **Graph Description:** The graph shows a rational function with two vertical asymptotes and no horizontal or slant asymptote visible in the given range. The vertical asymptotes are marked with dashed red lines. 1. **Vertical Asymptotes:** - At \(x = -3\) - At \(x = 3\) 2. **Graph Behavior:** - The function approaches infinity as it nears \(x = -3\) from the left and approaches negative infinity as it approaches from the right. - Similarly, the function approaches negative infinity as it nears \(x = 3\) from the left and approaches infinity as it approaches from the right. - Between the asymptotes \(x = -3\) and \(x = 3\), the function has a distinct behavior, moving from negative to positive. The graph is likely a rational function with factors in the denominator that create the vertical asymptotes at \(x = -3\) and \(x = 3\). **Equation Input Box:** There is a box below the graph for inputting the equation of the function. Based on the graph and the location of the asymptotes, the function may have factors of \((x + 3)\) and \((x - 3)\) in its denominator. Consider starting with an assumption of form for the rational function and adjusting based on any other discernible points or behaviors in the original graph.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Prompt:** Write an equation for the function graphed below. **Graph Description:** The graph shows a rational function with two vertical asymptotes and no horizontal or slant asymptote visible in the given range. The vertical asymptotes are marked with dashed red lines. 1. **Vertical Asymptotes:** - At \(x = -3\) - At \(x = 3\) 2. **Graph Behavior:** - The function approaches infinity as it nears \(x = -3\) from the left and approaches negative infinity as it approaches from the right. - Similarly, the function approaches negative infinity as it nears \(x = 3\) from the left and approaches infinity as it approaches from the right. - Between the asymptotes \(x = -3\) and \(x = 3\), the function has a distinct behavior, moving from negative to positive. The graph is likely a rational function with factors in the denominator that create the vertical asymptotes at \(x = -3\) and \(x = 3\). **Equation Input Box:** There is a box below the graph for inputting the equation of the function. Based on the graph and the location of the asymptotes, the function may have factors of \((x + 3)\) and \((x - 3)\) in its denominator. Consider starting with an assumption of form for the rational function and adjusting based on any other discernible points or behaviors in the original graph.
Transcribed Image Text:**Prompt:** Write an equation for the function graphed below. **Graph Description:** The graph shows a rational function with two vertical asymptotes and no horizontal or slant asymptote visible in the given range. The vertical asymptotes are marked with dashed red lines. 1. **Vertical Asymptotes:** - At \(x = -3\) - At \(x = 3\) 2. **Graph Behavior:** - The function approaches infinity as it nears \(x = -3\) from the left and approaches negative infinity as it approaches from the right. - Similarly, the function approaches negative infinity as it nears \(x = 3\) from the left and approaches infinity as it approaches from the right. - Between the asymptotes \(x = -3\) and \(x = 3\), the function has a distinct behavior, moving from negative to positive. The graph is likely a rational function with factors in the denominator that create the vertical asymptotes at \(x = -3\) and \(x = 3\). **Equation Input Box:** There is a box below the graph for inputting the equation of the function. Based on the graph and the location of the asymptotes, the function may have factors of \((x + 3)\) and \((x - 3)\) in its denominator. Consider starting with an assumption of form for the rational function and adjusting based on any other discernible points or behaviors in the original graph.
Expert Solution
Step 1

From the graph it is observed that the function has vertical asymptotes x=-1 and x=3.

Thus, the function is not defined at the point -1 and 3.

Also, the function has x intercept at the point 2,0 and y intercept at the point 0,-1.

Since, the function is not defined at the points  -1 and 3, the denominator of the function will be x+1x-3.

Therefore, the function is of the form y=ax+bx+1x-3.

Step 2

Now find the values of a and b using the intercepts as follows.

The y intercept at the point 0,-1. Therefore, when x=0 the value of y=-1.

-1=a(0)+b0+10-3a(0)+b0+10-3=-1b1(-3)=-1b-3=-1b=-1(-3)=3

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